Elliptic operators on refined Sobolev scales on vector bundles

We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product H\"ormander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate H\"ormander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.Comment: 22 page

A study of the conditions necessary for the onset of mid-latitude spread F

Ionospheric conditions associated with the initiation of spread F in the mid-latitude ionosphere were observed. The morphology of spread F at Puerto Rico was investigated. Data from 7 nights was examined for Arecibo, five with spread F and two without. The relative height of the F layer maximum and the vertically integreted Pedersen conductivity, the relation between E and F region conductivities, the coupling lengths between the E and F regions, and vertical and horizontal gradients of electron density were examined. At Millstone Hill 13 nights were examined for all of which spread F was observed. The EW and NS velocities and the vertical velocities and the electric ion temperature ratio were examined

The Hilbert Transform of a Measure

Let \fre be a homogeneous subset of \bbR in the sense of Carleson. Let $\mu$ be a finite positive measure on \bbR and $H_\mu(x)$ its Hilbert transform. We prove that if \lim_{t\to\infty} t \abs{\fre\cap\{x\mid\abs{H_\mu(x)}>t\}}=0, then \mu_s(\fre)=0, where \mu_\s is the singular part of $\mu$.Comment: 18 page